151012(5)

Ruby

Number of times k is used in writing out all the numbers 1 through n(2)

f(m, k, 10 ** i) をもう少し調べてみた。

# 高速化（ただし、0 < k ≦ m - 1）
def g(m, k, str, s)
  if s == 1
    return 0 if str.to_i < k
    return 1
  end
  a = str[0].to_i
  str1 = str[1..-1]
  if a < k
    b = 0
  elsif a == k
    b = 1 + str1.to_i(m)
  else
    b = m ** (s - 1)
  end
  return a * (s - 1) * m ** (s - 2) + b + g(m, k, str1, s - 1)
end

def f(m, k, n)
  str = n.to_s(m)
  g(m, k, str, str.size)
end

def f_ary(m, k, ary)
  ary.map{|i| f(m, k, i)}
end

ary = (0..30).map{|i| 10 ** i}
(2..10).each{|m|
  (1..m - 1).each{|k|
    p [m, k]
    p f_ary(m, k, ary)
  }
}

出力結果
[2, 1]
[1, 17, 319, 4938, 64613, 815030, 9884999, 114434632, 1314447116, 14846928141, 164293127179, 1809725656079, 19809942118413, 214309466746894, 2306412649693201, 24784747400675348, 264286863212871700, 2804216299269586964, 29761222783429247000, 314231954224804790291, 3301678091638143975450, 34722183171345671520285, 364370351327204996022297, 3799630573079809501429787, 39695385437637887626575902, 414481926914340004642684947, 4298429306008723825539153951, 44678554212330354478835302433, 464758015834485209941739044893, 4798702669950589113263389147172, 49657738936018521227027139264549]
[3, 1]
[1, 9, 150, 2194, 30374, 379432, 4508710, 51180200, 572158742, 6417405502, 70402455152, 769364418852, 8452692931320, 91673681872620, 993534122918024, 10779360089419044, 114661242594171254, 1216758202087170334, 12767569182895088152, 133862436956018539886, 1405771200630725286006, 14663860400505783919608, 153796494936787179989436, 1614986483014523055902106, 16887592497038459726705624, 177308495497250974401699424, 1846007098588454905102196728, 19172094511757798531897719532, 197677440096181467378574033316, 2037819660431890371250386506764, 21087982689937490731566483680876]
[3, 2]
[0, 6, 122, 1890, 24938, 319707, 3913633, 46711767, 541645055, 6121406962, 69078651468, 760016941672, 8206438460788, 88932763128457, 952772984410917, 10167982043918000, 108915998229254064, 1166408841920674213, 12364777379750241882, 131010763441671294609, 1385604273415163621734, 14644474302933408993470, 152454118578987694809569, 1587621877395315117388434, 16536992158262958640590360, 171612576527100685726563644, 1786660187871537792135309983, 18595499196279626763671447241, 193485155414562876382901957351, 2009058037618570051241478535319, 20801879171474381920146385631447]
[4, 1]
[1, 7, 119, 1270, 19482, 242178, 2520434, 31973411, 366171397, 3790564097, 44245876996, 488164976133, 5056725669893, 56692742610948, 610964919316485, 6310200205451268, 69321688173772803, 732713250404499465, 7571981434117226501, 82037556623093661699, 854087328285483073545, 8814800381086001004552, 94722289936057215483909, 973215603996092086091784, 10042775926285679481323524, 107193424611029871262433293, 1094927512861631153262559239, 11294729851272492427228741634, 119893894654631361202345738248, 1216129526537655672735115771912, 12564494019293974745563833303049]
[4, 2]
[0, 6, 70, 1260, 16409, 192212, 2519665, 28590949, 316981761, 3780691462, 40955973891, 442578674180, 5012229937156, 53342413107206, 568083953840132, 6253853871202310, 65744857114017797, 691251639160799237, 7504413864019427335, 78097532626468995080, 816632322249796878341, 8759273688423124172807, 90277557077891309633546, 942335136382397704044551, 10025010846995401075064838, 102648822381928313568362502, 1068615263197984839258079238, 11293727338714134579058311175, 115019501931404131503374860295, 1194374757940228516716625264650, 12549634416971078046754162081800]
[4, 3]
[0, 2, 65, 1204, 14361, 190320, 2422450, 26935136, 315646979, 3637836291, 39545638146, 439491002883, 4870493255682, 52137155514370, 563681888268292, 6110346662010885, 64610158962540550, 690125704852144131, 7342413742646296582, 77048432487621066756, 815479220551432011782, 8574054550918273171463, 89685252156628235452421, 942039916350659855646726, 9813799332178403535093770, 102319839960690909905944576, 1067443264974553916509257737, 11045048511171863736274124812, 114922309624224858618009223175, 1194099192736352461905824055305, 12271805249876734217354571939850]
[5, 1]
[1, 7, 65, 1226, 13001, 165627, 2028125, 20171876, 268203127, 2645703125, 32294921876, 331865234377, 3689306640628, 40104003906250, 403492431640629, 4679504394531250, 46297943115234377, 525622406005859377, 5326352691650390626, 57214763641357421878, 600465202331542968751, 6073609542846679687504, 66583832740783691406250, 664710590839385986328129, 7260833132266998291015627, 73223027288913726806640626, 775132898688316345214843753, 7986859233677387237548828126, 80951736219227313995361328129, 863699755929410457611083984377, 8654589133337140083312988281252]
[5, 2]
[0, 3, 65, 850, 13000, 134376, 1840627, 19937500, 221328126, 2637890627, 28398437500, 328398437500, 3360693359376, 38648925781253, 397962646484375, 4234832763671878, 46295654296875001, 482975616455078127, 5279622650146484376, 53592174530029296877, 587362003326416015628, 5973327350616455078127, 62607814311981201171878, 661748609542846679687501, 6841735947132110595703127, 72742526531219482421875006, 734739064872264862060546876, 7895457209646701812744140627, 79807786367833614349365234378, 827261740826070308685302734379, 8609556240960955619812011718752]
[5, 3]
[0, 2, 65, 726, 10501, 134375, 1559375, 19859376, 218984375, 2633984375, 27412109376, 306279296877, 3336279296876, 35572753906252, 397962646484375, 4179687500000002, 46295501708984375, 475630035400390627, 5086095428466796878, 53411624908447265628, 557347488403320312503, 5973314476013183593750, 61840394973754882812502, 661399910449981689453127, 6771569001674652099609381, 70993635952472686767578126, 734738297462463378906250001, 7598587559163570404052734380, 79595554493367671966552734377, 818210961408913135528564453127, 8609467113390564918518066406252]
[5, 4]
[0, 2, 41, 725, 9875, 134375, 1543751, 19859375, 218984375, 2413281251, 27158203127, 297734375001, 3330175781251, 35488525390625, 397956542968751, 4179595947265625, 44347412109375003, 470804443359375002, 4968225097656250002, 53234470367431640626, 554197692871093750000, 5967486095428466796876, 61696020603179931640627, 644959053993225097656253, 6700875508785247802734376, 69713067412376403808593751, 732756934762001037597656256, 7532236577570438385009765628, 79595398962497711181640625002, 818076492212712764739990234379, 8471354159526526927947998046882]
[6, 1]
[1, 7, 71, 731, 11198, 145691, 1471583, 15044020, 209959005, 2215950297, 22357576982, 260732117894, 2964571630298, 29734482337166, 300804840123645, 3596403661788849, 37220596165405190, 373942198480119148, 4097855055721481385, 44696639571507276727, 447357911583038454820, 4507221089612289882735, 50957678725204529674292, 522423493113073904971028, 5237942923211234920118775, 55861845725776606242564250, 597738819735637161095723198, 5978498177859560689393554348, 60114808605080898875434217157, 659222499890635670519267561691, 6722194098228926259163478574015]
[6, 2]
[0, 2, 64, 731, 8015, 105724, 1468480, 15036280, 170401845, 2126695994, 22349164466, 230521001428, 2553734780882, 29686729820416, 300655135938110, 3205679939605408, 36193990212721421, 373397802196303303, 3808925198520288348, 40481495552137836674, 447357910133301379941, 4506949179057162814305, 47098247702572249166961, 511150472996604417121406, 5237921416451673325535601, 53086650812786108689970093, 554300469172884983148693418, 5975675931659730378512521836, 60110711489062580279483779095, 621101636832399682997748155321, 6608209287937450403786033573638]
[6, 3]
[0, 2, 35, 724, 8015, 99035, 1317036, 15036215, 169406949, 1844445489, 22150503153, 230377772246, 2492643173066, 28053573741924, 300332901443709, 3189692886768142, 33445910414365065, 370957646611641398, 3797461074214285846, 39955008728061630146, 429356627553235346921, 4506931335876576179484, 46855356118085672416370, 484840463353143769132156, 5206242491098553003089692, 52914044231686680108760939, 548919206520818064857594777, 5784519881662083425823771238, 60105516962177037905394679990, 617957240997791333903390950104, 6350842080307719441451087214751]
[6, 4]
[0, 2, 34, 615, 7863, 99034, 1135666, 15005175, 160043362, 1814212292, 21038871313, 228282586508, 2486112825675, 26380129900105, 300261067622732, 3103725613575301, 33085595600958522, 359974948542551848, 3782714438353511379, 39826535380747513568, 414043307158573756000, 4506927171526374932207, 45978714519912057008219, 480312952883968590039808, 5095703161752445949084183, 52862369054840906489032458, 548402468946989137626886645, 5642867165502977095645810854, 60077582327775786557191105797, 609361800623766461960919687325, 6298555665517553486725726717854]
[6, 5]
[0, 1, 28, 472, 7792, 97934, 1135660, 14834162, 159266390, 1781669805, 19745872648, 228140521135, 2478725570884, 26379988532354, 297133827668287, 3088047662406271, 32833259197782770, 347720046322636951, 3782714433980619137, 39820035926441414782, 414031923911160746654, 4459946390681058028934, 45870436554855255952915, 478528920012052785004142, 4979022733663356406618128, 52862356372487454895214343, 548402342304638126483743083, 5639937353283198835033037458, 59476572009997077882211232957, 609067330162014991061127968705, 6288167401956831075730859273239]
[7, 1]
[1, 6, 78, 780, 8110, 88500, 1219870, 15865600, 172539968, 1731378357, 17700998204, 189189870494, 2383125982820, 26133698711472, 261771343230968, 2684380968983890, 27311303300643582, 312355937793457211, 3463944584394735456, 35601953767056758691, 357533815140350476230, 3630857630320495382668, 37779610844512699452723, 425041907065853528823289, 4466458203133567081386951, 44618338243477807545984051, 452940874938696904182501635, 4603505594762070258781889621, 50198736040249262034751004601, 532922641777229881768673946230, 5403147555399084356218382285511]
[7, 2]
[0, 2, 32, 752, 8051, 88500, 984600, 11630400, 151004956, 1731277200, 17693302672, 184945218049, 2049485093130, 21884202483741, 261771115832026, 2654159875976645, 27310892224213832, 283308244555221459, 3022490448478019193, 33862717664619078382, 357524905452287316162, 3616096044778082900587, 37155361718610826104304, 390754201533875335193348, 4067256720922519257062602, 44615002147323622393703377, 452926620881024031796277066, 4587844591618134507244906377, 46877981882897401112520582002, 487039597815156850879822720337, 5266080005736790770523241933164]
[7, 3]
[0, 2, 28, 430, 8051, 88492, 974738, 11630383, 127478384, 1591931287, 17660643600, 184945215113, 1980278392686, 21377898645382, 228752947068295, 2651059378227450, 27310527974405944, 283106134049192629, 3019387088183460063, 31625362502476074870, 348049734186983053933, 3615136257368942300470, 36998957507338517202370, 387486491649978060196670, 3986792603742625997814037, 41735379124070902004799090, 449837181933079970582104805, 4577735618037681059390970224, 46802753335038656499854386178, 487010005604531183489215431665, 5018057105178001109563306330253]
[7, 4]
[0, 1, 28, 430, 6047, 88296, 965000, 11630347, 125480000, 1407624801, 17660299864, 184666031090, 1949167908510, 21377239972530, 228441196782410, 2487517888462580, 27310527973598975, 283106054946074662, 3011076142524411632, 31120620488588141543, 327962317935941536555, 3612131317529933102576, 36960930792918110573472, 380594594233042078452575, 3985834697584601982285263, 41395096385504451603020184, 436957019354299011196766828, 4577286361550304008838349551, 46801617430924682807097416459, 486881184665268733846506432674, 4987113481566397526370865404541]
[7, 5]
[0, 1, 28, 430, 5649, 87266, 965000, 10924285, 125480000, 1404549428, 15793795402, 184661800871, 1948922950960, 21300782832428, 228439874081111, 2417740139508408, 27310527942304340, 282144405942988784, 2972279068997955805, 31120608681853113819, 325686674703165126747, 3462287588768112743296, 36960543171334196001997, 379831996450583231248461, 3985670485000432091737956, 41340947588645677371268945, 427109540750639968151691298, 4577285016667239708602472127, 46799030528549538811232959217, 486368401788157409062497692827, 4961218635023759683623964907660]
[7, 6]
[0, 1, 28, 402, 5649, 70452, 965000, 10806308, 125476702, 1401843466, 15676594033, 184661795894, 1948882592551, 21280350842800, 228439219364426, 2417740139427705, 25905396891838509, 281446506360443052, 2929632209731661992, 31066695222334957036, 325685924657652990733, 3419989862347559869333, 36960543167667485287638, 379824176805007286224579, 3985669766703621617211638, 41329191803312980103824816, 427104965075225917353573209, 4472837222595409375898469390, 46799006370315540059932519877, 478621614493220845244505892843, 4961215723788366126299752391373]
[8, 1]
[1, 5, 66, 870, 9418, 95404, 1020178, 10910114, 116961408, 1269319104, 15420668802, 186335569154, 2092258707459, 22324231147523, 223533368487937, 2300102388453377, 23678250753204227, 244218428740583426, 2521303233634959362, 26987371554266808324, 307381704052157972481, 3312167967401039364099, 34662251320016253747201, 352132946710859256692738, 3580993929556662847275012, 36151550056345656205770756, 369998943087644997308121092, 3791006228343139484888465411, 38829724147853438538318086146, 428005188847297506828343050243, 4528454022449639996752494854147]
[8, 2]
[0, 2, 29, 381, 7124, 95373, 1020112, 10902625, 116961408, 1269319104, 13607942531, 155047253248, 1641958131714, 18648704704515, 217975554600962, 2300096106074114, 23675007796813825, 244187609128878083, 2521303226531643392, 26210740265861906435, 279711585926283001859, 2902447824095134089218, 30494440078365576134661, 338547063815532455133187, 3560961916499477187788801, 36125179996591085238353924, 369092248722919565681491969, 3791005696884324459807244289, 38733214357050152443388624898, 398277592615671175209692430339, 4162275061228438708256216449026]
[8, 3]
[0, 1, 29, 381, 5075, 63918, 971537, 10902241, 116953473, 1262686145, 13539799424, 151967442178, 1641152821249, 17340401500161, 188344290390017, 2170897047359490, 23541908635230209, 243975985889525762, 2521303173503582210, 26194734692110041089, 271945273057034108929, 2901583130260266549252, 29939173041882216792066, 314049571512541934256131, 3333146630949991994097667, 36125178266081207951294465, 368828628825374481058889731, 3785179414964811973682790403, 38733213799036144213674688513, 397312795729494474951145553921, 4162259662471593636724647919620]
[8, 4]
[0, 1, 26, 381, 4755, 62053, 754449, 10416353, 116928577, 1252198850, 13523801473, 146455220993, 1635778152450, 17333126389762, 188343491354627, 1993331687067651, 22106987706359810, 242568603890958340, 2519252240709910530, 26192725864872542208, 271945272702129668099, 2895818520444168568834, 29931803566205061038083, 314030922173988236951554, 3236253103937606641516546, 34035718740737068781010949, 368828607838599311056175107, 3771578999494112416582598658, 38732979413418482214763495429, 397312795683377614743652007939, 4132037974362306941130373070854]
[8, 5]
[0, 1, 20, 374, 4738, 62052, 753872, 8804960, 116126274, 1250426306, 13523022208, 145523411715, 1632197982209, 17331810058241, 188287656763393, 1978494012395521, 21106490375577602, 233561403058503682, 2519039379033489411, 26104662699562631169, 271929331982013300738, 2858925012230660161539, 29929803302553834225665, 310138859550804502642690, 3236251086324973562757121, 33704710365076977753260034, 352771673583441351882571780, 3770361747262021455231057923, 38713870975677183393735376897, 397312795351332306645457829892, 4073127952114241036450256650241]
[8, 6]
[0, 1, 20, 373, 4738, 62052, 738065, 8581345, 99885377, 1250328001, 13523019137, 145381182209, 1601405499393, 17331800784897, 186638389207042, 1978494010724353, 21105998742265857, 223633395596869632, 2510204908743884802, 26048610146534424577, 271496986414564507650, 2818155688920055808001, 29929803278063834234882, 309308555691062882992130, 3235699557423071344197638, 33701940171669964601688064, 349484656471512754788761603, 3685703883323617320787509251, 38713867991782628793121767426, 397194394407746599290968997892, 4073025465266024431629950255109]
[8, 7]
[0, 1, 20, 350, 4738, 62052, 721104, 8542816, 99221825, 1176403393, 13523010945, 145381180160, 1563478173696, 17331793272832, 183339854176257, 1978482359373826, 21105940483448834, 223633362806456321, 2366089719996940290, 26048367154463113218, 270632115350116499459, 2818140398861121486850, 29894765705872778199042, 309308555623536970956802, 3235660646082689049296897, 33701940171669964601688064, 349446529532967592010973185, 3614157732169292206051426307, 38713397461057074648068390916, 397155707275698684902185631747, 4072715239424805790426115080194]
[9, 1]
[1, 4, 50, 663, 8260, 99109, 1143090, 12527839, 135246289, 1409108421, 14660616984, 155444059591, 1630739921525, 17024720710829, 180215825517231, 1876739468499213, 19764853172536251, 205589163931606489, 2155867004951010588, 22558669798425164931, 235436287048196113944, 2460489043375692066252, 26752599254052604541911, 294272850260761602792780, 3112641278712249425091604, 32764467158341991897094435, 344095897737184879884337351, 3596843315781575484981559027, 36918489296419678748627212344, 379144290872420566267722670492, 3844334443631408268993122982013]
[9, 2]
[0, 1, 22, 390, 4819, 57508, 674529, 8178624, 105633660, 1242423463, 14199075590, 153071555631, 1630715025092, 17024720656758, 180215824612925, 1876739161047531, 19764535935738235, 205586904471988091, 2155866990727562950, 22558667365361933382, 234107582597019724440, 2441125857514994917013, 25420374302574773229657, 267236552503520584934116, 2830531561546103452320004, 29225200535877994572766137, 302218510988585783455384456, 3115197559749843710226384131, 32731708232200714434912562688, 351487752411994433553171503806, 3699655842625692803369545185276]
[9, 3]
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[9, 5]
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[9, 7]
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[10, 1]
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[10, 3]
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[10, 4]
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[10, 5]
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[10, 6]
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[10, 7]
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[10, 8]
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[10, 9]
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